We can use 'may' to ask for permission. However this is rather formal and not used very often in modern spoken English

May I borrow your pen?

May we think about it?

May I go now?

We use 'may' to suggest something is possible

It may rain later today.

I may not have time to do it today.

Pete may come with us

might

We use 'might' to suggest a small possibility of something. Often we read that 'might' suggests a smaller possibility that 'may', there is in fact little difference and 'might is more usual than 'may' in spoken English.

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